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A353055
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Successive records of function f(x) = log(abs(pi(x) - R(x)))/log(x) where pi(x) is the number of primes <= x and R(x) is Riemann's prime counting function.
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0
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2, 4, 7, 10, 19, 47, 58, 73, 109, 113, 1109, 1123, 1129, 1307, 1321, 1327, 1418, 1419, 1420, 1421, 1422, 5379, 5380, 7449, 7450, 10343, 11676, 11761, 11762, 11763, 11764, 11765, 11766, 11767, 11768, 11769, 11770, 11771, 11772, 11773, 11774, 11775, 11776, 19360, 19361, 19362, 19363, 19364, 19365, 19366, 19367, 19368, 19369, 19370, 19371, 19372
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OFFSET
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1,1
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COMMENTS
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Two possibilities:
(1) this sequence is finite;
(2) this sequence is infinite.
In case (1) there exists a maximal integer x_max such that J = f(x_max) = log(abs(pi(x_max) - R(x_max)))/log(x_max).
In case (2) there exists a real constant J such that lim_{x->oo} f(x) = J.
Then for every positive integer x, abs((R(x) - pi(x))/x^J) <= 1.
According to actual computations biggest x = 1090696 with log(-85020 + R(1090696))/log(1090696]) = 0.27835121240340474... and no more new terms up to x 3000000. Follow this:
0.27835121240340474... <= J.
J < 1/2 = limit((log(x) - 2*log((8*Pi)/log(x)))/(2*log(x)), x -> infinity) proof follow Lowell Schoenfeld 1976 proof on upper limit of Chebyshev function psi(x).
Constant J can be used to measure best proved upper limits of asymptotical behavior of function pi(x) when x->infinity. If J is smaller upper bound is better.
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LINKS
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EXAMPLE
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x f(x) comment
1 -infinity a(1)
2 -0.8862754573970588 a(2)
3 -4.883591467412115 removed because f(3) < f(2)
4 -0.614424415865155 a(3)
5 -1.0695141714266385 removed because f(5) < f(4)
... ...
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MATHEMATICA
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gg = {1}; imax = -1000; Do[
kk = Log[Abs[PrimePi[x] - RiemannR[x]]]/Log[x];
If[kk > imax, AppendTo[gg, x]; imax = kk], {x, 2, 20000}]; gg
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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